+lemma cpt_inv_sort_sn (h) (n) (G) (L) (s):
+ ∀X2. ⦃G,L⦄ ⊢ ⋆s ⬆[h,n] X2 →
+ ∧∧ X2 = ⋆(((next h)^n) s) & n ≤ 1.
+#h #n #G #L #s #X2 * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
+/2 width=1 by conj/
+qed-.
+
+lemma cpt_inv_zero_sn (h) (n) (G) (L):
+ ∀X2. ⦃G,L⦄ ⊢ #0 ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #0 & n = 0
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,n] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓓV1
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬆[h,m] V2 & ⇧*[1] V2 ≘ X2 & L = K.ⓛV1 & n = ↑m.
+#h #n #G #L #X2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ elim (ist_inv_plus_SO_dx … H2) -H2 // #m #Hc #H destruct
+ /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
+]
+qed-.
+
+lemma cpt_inv_zero_sn_unit (h) (n) (I) (K) (G):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ⬆[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+#h #n #I #G #K #X2 #H
+elim (cpt_inv_zero_sn … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpt_inv_lref_sn (h) (n) (G) (L) (i):
+ ∀X2. ⦃G,L⦄ ⊢ #↑i ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #(↑i) & n = 0
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬆[h,n] T & ⇧*[1] T ≘ X2 & L = K.ⓘ{I}.
+#h #n #G #L #i #X2 * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
+| #I #K #V2 #HV2 #HVT2 #H destruct
+ /4 width=6 by ex3_3_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma cpt_inv_lref_sn_ctop (n) (h) (G) (i):
+ ∀X2. ⦃G,⋆⦄ ⊢ #i ⬆[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G * [| #i ] #X2 #H
+[ elim (cpt_inv_zero_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpt_inv_lref_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cpt_inv_gref_sn (h) (n) (G) (L) (l):
+ ∀X2. ⦃G,L⦄ ⊢ §l ⬆[h,n] X2 → ∧∧ X2 = §l & n = 0.
+#h #n #G #L #l #X2 * #c #Hc #H elim (cpg_inv_gref1 … H) -H
+#H1 #H2 destruct /2 width=1 by conj/
+qed-.
+